By Albert Nijenhuis, Herbert S. Wilf

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**Extra resources for Combinatorial Algorithms for Computers and Calculators, Second Edition (Computer Science and Applied Mathematics) **

**Sample text**

The choices between loss vs. no-loss and gain vs. no-gain are made to minimize the associated costs. 2 Applications Parsimony’s simple assumptions are appreciated even in contemporary studies of complex genome features. A case in point is Wagner parsimony that was recently used to study genome size evolution [6] and short sequence length polymorphisms [51]. Genome size and tandem repeat copy numbers as well as the other examples to follow are common in that they are difficult to address in probabilistic models, either for technical reasons or simply because the relevant evolutionary processes are still not understood well enough.

A node u ∈ V and all its descendants form the subtree rooted at u, denoted by Ψu . Every node u ∈ V is associated with a label ξ [u] ∈ F over some feature alphabet F . The labels ξ [x] represent the states of a homologous character at different nodes of the phylogeny. Labels are observed at the terminal nodes, but not at the other nodes, which represent hypothetical ancestors; see Fig. 1. We state the problem of ancestral reconstruction in an optimization setting, where the label space is equipped with a cost function d : F × F → [0, ∞].

Classically, the placement is considered in k-dimensional Euclidean space with F = Rk , and d is the ordinary Euclidean distance. General parsimony labeling gives the optimal placement of Steiner vertices for a fixed topology. The algorithmic difficulty of parsimony labeling depends primarily on the assumed cost function d. Finding the most parsimonious tree is NP-hard under all traditional parsimony variants [14–16], but computing the score of a phylogeny is not always difficult. 2 A Quick Tour of Parsimony Variants The minimum total change of Eq.